I have an old oil furnace and I am looking for some clarification as to how it is supposed to work. It heats water filled radiators through out the house.

Currently it is connected to a thermostat by three wires. When it is needed the thermostat does heat the house but lets say it is 80 degrees outside, the radiators will not get hot but the furnace will still come on periodically for a short period of time. Is this how this system is supposed to run or might there be something wrong with the thermostat / wiring.

Note: The thermostat does manage the alert the furnace when it needs to but the air temperature is a bit different than the reading on the thermostat. ex thermostat says 65 and the actual temperature is 72.

Some equipment will run a periodic maintenance cycle so the equipment does not stagnate from lack of use. Though I've never seen this on residential equipment. These cycles are more on the order of weekly or monthly rather than daily. Could explain things, but unlikely so IMO. Thought it was worth mentioning all the same.
–
bcworkzOct 23 '13 at 2:57

In S-Plan, electric power only reaches the furnace (AKA boiler) after passing through a series of devices:

power from the supply is connected to a timer/programmer

during set times, the timer/programmer allows power to the room thermostat

when temperature is low, the room thermostat allows power to the zone valve

when the zone valve is fully open, it allows power to the pump and furnace/boiler

Some furnaces/boilers have an independent power supply and control the pump so that they can overrun to circulate water to remove residual heat from the furnace/boiler.

None of the domestic systems I've had in my homes have ever needed to start the furnace/boiler unless the thermostat is calling for heat. I changed my room thermostat to a model with an indicator light that comes on when it is calling for heat.

The system described above is a high-voltage system, I believe many US systems use a completely different low-voltage control arrangement.

Most older furnaces are very simple: when the temperature drops below a set point the furnace comes on; when it rises above a different temperature (usually a degree or two higher) it goes off. The furnace probably shouldn't be coming on if it's 80 degrees, but if you're setting the temperature that high for testing it may just not be running long enough to warm up.

I'm not sure what wires you have and what they do; typically there would be two for a single-zone system. Additional wires could be used to control a fancy multi-stage blower, a humidifier or dehumidifier, an AC unit, extra power for a high-tech thermostat etc. What color are the wires? There are conventions that tell you what the colors should mean (although of course there's nothing that guarantees the thermostat was wired conventionally).

As for the thermometer being off: there may be an adjustment screw to help you calibrate the thermostat. Check the back or sides, or look up a manual.